On the number of connected components in the space of closed nondegenerate curves on 𝑆_{𝑛}

Abstract

The main definition. A parametrized curve γ : I → R is called nondegenerate if for any t ∈ I the vectors γ′(t), . . . , γ(t) are linearly independent. Analogously γ : I → S is called nondegenerate if for any t ∈ I the covariant derivatives γ′(t), . . . , γ(t) span the tangent hyperplane to S at the point γ(t) ( compare with the notion of n-freedom in [G]). Fixing an orientation in R or S we call a nondegenerate curve γ right-oriented if the orientations of γ′, . . . , γ coincide with the given one and left-oriented otherwise. Nondegenerate curves on S are closely related with linear ordinary differential equations of (n + 1)-th order. Such an equation defines two nondegenerate curves on S ⊂ V (n+1)∗, where V (n+1)∗ is the (n + 1)-dimensional vector space dual to the space of its solutions as follows. For each moment t ∈ I we choose the linear hyperplane in V n+1 of all solutions vanishing at t i.e. obtain a unique curve in the projective space P. Raising it to S we obtain a pair of curves; both of them are right-oriented if n is odd and have opposite orientations if n is even (nondegeneracy follows from nonvanishing of its Wronskian). A nondegenerate curve γ : [0, 1] → S defines a monodromy operator M ∈ GLn+1 which maps γ(0), γ ′(0), . . . , γ(0) to γ(1), γ′(1), . . . , γ(1). In the paper [K-O] there is given the complete set of invariants for symplectic leaves of the second Gelfand-Dikii bracket; namely its leaves are enumerated by pairs consisting of 1) monodromy operator, and 2) the connected component of the space of right-oriented curves in the sphere with the given monodromy operator. In this paper we study the number of connected components for closed nondegenerate right-oriented curves (corresponding to the identity monodromy operator). Nondegeneracy is also interesting in connection with the general theory of the hprinciple (see [G]). Let NR (NS) be the space of all nondegenerate closed right-oriented curves in R(S respectively). The question we consider is how to calculate πo(NS) and πo(NR). The first paper studying a similar question is [F]. Later J.Little [L1,L2] studied NS and NR and proved the following (W.Pohl’ conjecture): card(πo(NS)) = 3 and card(πo(NR)) = 2. (The invariant which distinguishes closed nondegenerate curves is an element of π1 of the image of the natural map ν : NR → SOn, where